3 Last Lecture We talked about raster data as one form of tesselation, their properties and important things you will find when working with them These facts are important to understand storage limitations, data types, bit depths and resolution-related problems In this context the assignment of pixel values in classifications or vector-raster conversions can become complex and is crucial

4 Last Class Meeting Right, and remember the last session which was a readings discussion about the field-object debate Take some impressions with you from this nice session - we did not come to an end (nobody ever did) Keep in mind the different conceptual model approaches and their counterparts in different disciplines Try to remember what the advantages of field representations are in modeling, error analysis and mapping

5 Today s Outline We will talk about some important basics of raster analysis and Map Algebra You will hear about the principles of Map Algebra as the foundation of GIS modeling framework We will talk about the single building blocks of this modeling language: operators, functions, parameters and objects This will give you an impression of the concepts behind it

6 Learning Objectives You will learn what Map Algebra stands for and what the single components are You will have insights into the functioning of Map Algebra You will understand what functions you are going to use, what an operator is and how to combine all elements for iterative models under control

7 Some Repitition

8

9 World Files for Georeferencing Information Some image formats store GI in a header of the image file (grids, img, GeoTIFF) Others use world (ASCII) files (.tfw) Origin of an image is ul (row values increase downward), of a coord system ll

11 Raster and Raster Analysis Two-dimensional arrays organized in columns and rows as basis for efficient computation (translation into code) Making use of the simple and flexible data structure Fields, objects, regions, connected components, networks Developing and extending functionalities and operators MyRaster[row][col]

12 Understanding Raster Analysis To apply analysis tools is easy and straightforward Behind existing tools are complex algorithms implying mathematics, geometry and matrix operations between datasets and within regions of the same layer Thus a basic understanding of the concepts behind these tools is fundamental

15 Map Algebra and Matrix Algebra One-to-one locational correspondence throughout all functions applied *, /, **, root are defined by the same rules of maintaining the one-to-one translation Matrix Algebra would apply rules for mathematical matrices (do you remember them?) If A is a 2x3 and B is a 3x2 matrix:

16 Why we are not doing Matrix Algebra So that is the special sense of Map Algebra: Position of individual grid cells corresponds to their position in geographic space (not in math matrices) Cell values are changed but not their position Intuitively easy to get but it is important to know these differences and why they exist

18 Map Algebra as Modeling Language Something like a modified version of Matrix Algebra, yes! But it can be seen as a complete modeling language (taken as standard for industry) Allows for program control, development and iteration + mathematical manipulation and logical operators of comparison and thus for the whole complexity of modeling Fuzzy logic as one example

25 Zonal Functions Zones identified from another layer for evaluation of the target cell Zones are geographic areas with certain characteristics (not necessarily connected such as regiongroups) Internal attribute homogeneity meangrid = zonalmean(zonalgrid,valuegrid) Statistics are written into each cell of a zone Blocks similar but predefined roving window corresponds to the zone

26 Global Functions Operate on the entire grid at once Each cell is a function of all input cells EucDistance, WeightDistance, Hydrological, Surface, Visibility, Viewshed, Max, eucdist( )

27 Further Functions We will see many specialized functions implemented in GIS tools Complex functions which integrate many of basic functions Hydrological functions Surface analysis

28 Flow Control Integral component of Map Algebra Command line framework / GUI as interface with the GIS user Composed of two elements (that work with operations and functions): Statements: verbal representation of operations to link operators, functions and programming commands Programs: notational representation of a procedure in Map Algebra; ordered sequence of statements

29 Iteration Repeated execution of the same sequence of statements under varying conditions or with other datasets or subsets or based on testing for conditions ( for looping, while looping ) Combined with if then logic Makes a modeling framework more powerful

30 ... Geoprocessing Framework in ArcGIS Tool dialog: Single tasks Command line Model-building: This is GIS2! Scripting/Programming: This is where we ll go in GIS3!

31 Graphical Modeling Framework Different Buffer distances for different features in different areas

33 More on Local Functions Basically the operators we have seen before are executed cell-by-cell Remember the handling with NoData values, 0-values and values unequal to zero Higher level operations based on local processing are reclassifications, nested functions and overlays trigonometric, exp., log., select, statistical,

34 Local Functions: Reclassifications Assigning output values that depend on the specific set of input values Based on a table, ranges of values (for automated reclassification) or conditional tests

36 Local Functions: Overlay I Cell-by-cell comparison to register unique combinations of variables vectors of attributes given in a table (many to one relationships and extended rasters)

37 Local Functions: Overlay II Multiplication to mimic clip (extraction) using source and template layers (binary masks) Addition to mimic union (same id s for disjunct pixels with same characteristics through many-to-one rs. + ambiguous if same values out of different combinations) Combining zonal with overlay

38 More on Neighborhood Functions Often based on the concept of moving windows: Configuration of raster cells that is positioned over the input raster and defines the input for an operation to be applied. Result associated with center and written to the output. Window moves to the next location Much depends on the neighborhood Any ideas of functions that use NF?

42 The Principles of Moving Windows Left-to-right and top-to-bottom Dimensions: size of the neighborhood Odd-numbered in x and y to provide a natural center cell and square- or rectangular-shaped (or L or wedge or circular)

43 Examples for Moving Window NFs

44 Moving Windows and Kernels Set of constants for a given window size and shape Are applied with a function to every moving window location What can you see at the margins of the output grid?

50 Moving Windows and Spatial Covariance The more moving window operations / functions carried out the more related (autocorrelated) the cell values are Adjacent cells share six of nine cells in the local neighborhood for the computation

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